Does the following integral have a closed form? $$ \mathcal{J}(2,3,5,7,11) = \int_0^\infty x J_0(x\sqrt{2})J_0(x\sqrt{3})J_0(x\sqrt{5})J_0(x\sqrt{7})J_0(x\sqrt{11})\,dx. $$
I know that some similar integrals do have a closed form $$ \mathcal{J}(2,3,5) = \frac{1}{\pi \sqrt{6}} \approx 0.12995. $$ $$ \mathcal{J}(2,3,5,7) = \frac{1}{\pi^2 210^{1/4}}K(\tfrac14\sqrt\alpha) \approx 0.110411, \qquad 43-672\alpha+42\alpha^2 = 0, \qquad \alpha\approx 15.9358. $$ (Here $K(k)$ is the complete elliptic integral of the first kind with modulus $k$.)
Numerically, the integral is approximately $$ 0.061064349908721692\ldots, $$ but I could not find a symbolic value for this constant.